reserve x,y,y1,y2,z,e,s for set;
reserve alpha,beta,gamma for Ordinal;
reserve n,m,k for Nat;
reserve g,g0,g1,g2,gO,gL,gR,gLL,gLR,gRL,gRR for ConwayGame;

theorem Th48:
  g is positive iff
    (for gR st gR in the_RightOptions_of g holds gR is fuzzy or gR is positive)
    & (ex gL st gL in the_LeftOptions_of g & gL is nonnegative)
proof
  hereby
    assume
A1:   g is positive;
    hence
A2:   for gR st gR in the_RightOptions_of g holds
        gR is fuzzy or gR is positive by Th45;
    consider gL such that
A3:   gL in the_LeftOptions_of g & gL is non fuzzy non negative
      by Th47,A1,A2;
    take gL;
    thus gL in the_LeftOptions_of g & gL is nonnegative by A3;
  end;
  hereby
    assume
     for gR st gR in the_RightOptions_of g holds gR is fuzzy or gR is positive;
    then
A4:   g is nonnegative by Th45;
    assume ex gL st gL in the_LeftOptions_of g & gL is nonnegative;
    then consider gL such that
A5:   gL in the_LeftOptions_of g & gL is nonnegative;
    gL is non fuzzy non negative by A5;
    then not g is zero by Th47,A5;
    hence g is positive by A4;
  end;
end;
